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This article is cited in 2 scientific papers (total in 2 papers)
Higher Moments of the Error Term in the Divisor Problem
A. Ivića, W. Zhaib a University of Belgrade
b Shandong Normal University
Abstract:
It is proved that, if $k\ge 2$ is a fixed integer and $1\ll H\le(1/2)X$, then
$$
\int_{X-H}^{X+H}\Delta^4_k(x)\,dx \ll_\varepsilon X^\varepsilon (HX^{(2k-2)/k}+H^{(2k-3)/(2k+1)}X^{(8k-8)/(2k+1)}),
$$
where $\Delta_k(x)$ is the error term in the general Dirichlet divisor problem. The proof uses a Voronoï–type formula for $\Delta_k(x)$, and the bound of Robert–Sargos for the number of integers when the difference of four $k$th roots is small. The size of the error term in the asymptotic formula for the $m$th moment of $\Delta_2(x)$ is also investigated.
Keywords:
Dirichlet divisor problem, higher moments, mean fourth power, Voronoï formula, Euler's constant $\gamma$, residue theorem.
Received: 21.04.2009 Revised: 18.02.2010
Citation:
A. Ivić, W. Zhai, “Higher Moments of the Error Term in the Divisor Problem”, Mat. Zametki, 88:3 (2010), 374–383; Math. Notes, 88:3 (2010), 338–346
Linking options:
https://www.mathnet.ru/eng/mzm8810https://doi.org/10.4213/mzm8810 https://www.mathnet.ru/eng/mzm/v88/i3/p374
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